Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces

The stolen necklace problem asks for a guaranteed way to split a line of jewels between two thieves so that each person gets exactly half of every jewel type—while using as few cut points as possible. The central claim is that with N different jewel types, a fair division can always be achieved using no more than N cuts, no matter how the jewels are arranged. Proving that kind of “always possible” statement is notoriously difficult in discrete optimization, so the breakthrough comes from translating the discrete puzzle into a continuous geometric setting where a powerful topological theorem forces a solution to exist.

The key topological tool is the Borsuk-Ulam theorem. It concerns continuous maps from a sphere to a lower-dimensional space: if a sphere is continuously “squished” into a plane (or, more generally, into fewer dimensions), then some pair of antipodal points—points directly opposite each other on the sphere—must land on the same output. A classic intuition comes from Earth: if temperature and pressure vary continuously over the globe, then there must be two opposite points where both temperature and pressure match exactly. The proof idea in the transcript reframes the problem by building a new function that measures the difference between outputs at antipodal points; continuity then forces that difference to hit the origin, which means the antipodal outputs coincide.

To connect this to stolen necklaces, the transcript “continuousifies” the discrete jewel arrangement. For the two-jewel case (say sapphires and emeralds), the necklace is treated as a unit interval divided into equal subsegments, each segment colored by its jewel type. A fair division using two cuts corresponds to choosing cut locations that determine how much total colored length each thief receives. Crucially, any fair division in the continuous model can be adjusted so the cuts align with the original discrete segment boundaries, so solving the continuous version suffices.

Then comes the geometric encoding: all possible ways to choose the relative lengths of three pieces (for two cuts) are represented by points on a sphere. A point (x, y, z) on the unit sphere satisfies x² + y² + z² = 1, and the squares x², y², z² become the lengths of the three necklace pieces. The sign of each coordinate decides which thief receives that piece: positive sends it to thief 1, negative sends it to thief 2. Under this correspondence, moving to the antipodal point (−x, −y, −z) keeps the cut lengths the same (because squares don’t change) but swaps every piece’s owner—exactly the “antipodal swap” that should preserve fairness.

Finally, the fairness requirement becomes a statement about a continuous map from the sphere to a plane: take the outputs to be the total sapphire length (and emerald length) assigned to thief 1. If an antipodal pair of points on the sphere map to the same output, then swapping all pieces leaves those totals unchanged—meaning both thieves receive equal amounts of each jewel type. The Borsuk-Ulam theorem guarantees such an antipodal collision, so a fair division with two cuts exists for two jewel types. The general N-jewel case follows by using the higher-dimensional versions of Borsuk-Ulam on hyperspheres, where mapping a k-dimensional sphere into k−1 dimensions forces the same kind of antipodal coincidence.

In short: the “always possible with N cuts” guarantee for the stolen necklace problem is forced by topology once the discrete allocations are encoded as antipodal sign choices on a sphere (or hypersphere). The payoff is a rare bridge between discrete fairness constraints and continuous geometry, where existence is not guessed—it’s compelled.